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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 24 — Nov. 27, 2006
  • pp: 11631–11652
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Numerical calculations of ARROW structures by pseudospectral approach with Mur’s absorbing boundary conditions

Chia-Chien Huang  »View Author Affiliations


Optics Express, Vol. 14, Issue 24, pp. 11631-11652 (2006)
http://dx.doi.org/10.1364/OE.14.011631


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Abstract

The pseudospectral method, proposed in our previous work, has not yet been constructed for optical waveguides with leaky modes or anisotropic materials. Our present study focuses on antiresonant reflecting optical waveguides (ARROWS) made by anisotropic materials. In contrast to the fields in the outermost subdomain expanded by Laguerre-Gaussian functions for guided mode problems, the fields in the high-index outermost subdomain are expanded by the Chebyshev polynomials with Mur’s absorbing boundary condition (ABC). Accordingly, the traveling waves can leak freely out of the computational window, and the desirable properties of the pseudospectral scheme, i.e., provision of fast and accurate solutions, can be preserved. A number of numerical examples tested by the present approach are shown to be in good agreement with exact data and published results achieved by other numerical methods.

© 2006 Optical Society of America

1. Introduction

Building optical integrated circuits for practical use in an optical communication system requires a variety of photonic components such as optical interconnectors, modulators, switches, filters, splitters, polarizers, and couplers. The diverse optical waveguides by using distinct materials and refractive index profiles, according to specifications, produce functional optical integrated circuits. To precisely compute propagation characteristics (i.e., dispersion, attenuation, mode distribution, and birefringence) of general optical waveguides with both guided and leaky modes, we may precisely tailor the expected optical devices. A low-loss waveguide antiresonant reflecting optical waveguide (ARROW), which utilizes two thin interference cladding films made on semiconductor substrates [1

1. M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49, 13–15 (1986). [CrossRef]

], has been proposed to construct a wide range of functional devices [2–5

2. T. Baba, Y. Kokubun, T. Sakaki, and K. Iga, “Loss reduction of an ARROW waveguide in shorter wavelength and its stack configuration,” J. Lightwave Technol. 6, 1440–1445 (1988). [CrossRef]

]. Unlike total internal reflection mechanisms for guided modes, ARROW structure supports leaky waves through high reflectivity corresponding to the antiresonant condition of a Fabry-Perot resonator. This kind of waveguide has several advantages [6

6. T. Baba and Y. Kokubun, “Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides-numerical results and analytical expressions,” IEEE J. Quantum Electron. 28, 1689–1700 (1992). [CrossRef]

,7

7. W. P. Huang, R. M. Shubair, A. Nathan, and Y. L. Chow, “The modal characteristics of ARROW structures,” J. Lightwave Technol. 10, 1015–1022 (1992). [CrossRef]

] such as low loss, relatively large spot size for effectively coupling into optical fibers, high fabrication tolerance for thicknesses of interference claddings, high loss discrimination between the fundamental and higher-order modes, high polarization sensitivity, and polarization insensitivity [8

8. T. Baba and Y. Kokubun, “New polarization-insensitive antiresonant reflecting optical waveguide (ARROW-B),” IEEE Photon. Technol. Lett. 1, 232–234 (1989). [CrossRef]

].

2. Formulations of planar waveguides

Considering an anisotropic medium if the principal dielectric axes of the crystal are found to be parallel to the waveguide coordinate system, the dielectric tensor ε̿ is expressed as follows:

ε==ε0[nxx2000nyy2000nzz2],
(1)

x(Ey(x)x)+k02(nyy2(x)nff2)Ey(x)=0,
(2)

for transverse-electric (TE) and that

nxx2(x)x(1nzz2(x)Hy(x)x)+k02(nxx2(x)neff2)Hy(x)=0
(3)

for transverse-magnetic (TM) polarizations. Here, k 0=2π/λ 0, λ 0 denotes the wavelength in free space, neff denotes the complex effective refractive index, and Ey (x) and Hy (x) represent, respectively, the components of the electric and magnetic fields in the y -direction.

3. Numerical scheme

3.1 Pseudospectral scheme and interfacial patching conditions

The framework of the pseudospectral scheme is to divide computational domain into a number of subdomains determined by discontinuous material interfaces. In each subdomain, the optical field φ(x) is expanded by a set of proper interpolation functions Ck (x) and the unknown grid point values φk as follows:

φ(x)=k=0nCk(x)φk,
(4)

where

Ck(x)=ψn+1(x)ψn+1(x)(xxk),0kn.
(5)

Here, ψ n+1(x)denotes an arbitrary basis function of order n+1 and is determined according to the feature of that subdomain, the prime denotes the first derivative of ψ n+1(x) with respect to x, and xk denotes the corresponding collocation point fulfilling the condition of Cl (xk )=δlk , where δlk denotes the Kronecker delta. The explicit forms of Ck (x) for distinct basis functions are represented below. If we take the Chebyshev polynomials as basis functions, we have [19

19. J. P. Boyd, “Chebyshev and Fourier Spectral methods,” in Lecture Notes in Engineering, 2nd ed. (Springer Verlag, 2001).

]

Ck(x)=(1)k+1(1x2)Tn(x)ckn2(xxk),xxk
(6)

where Tn (x) denotes the Chebyshev polynomial of order n, c 0=cn =2, and ck =1 (1≤kn-1). For the LG functions, namely ψ n+1(αx)=exp(-αx/2)(αx)Ln (αx) whereLn (αx)denotes the Laguerre polynomial of order n, the explicit form ofCk (x)is given as follows [19

19. J. P. Boyd, “Chebyshev and Fourier Spectral methods,” in Lecture Notes in Engineering, 2nd ed. (Springer Verlag, 2001).

]:

Ck(αx)=eαx2eαxk2(αx)Ln(αx)(αxLn)(αxk)(αxαxk),xxk.
(7)

AΦ=(k0neff)2IΦ,
(8)

where I denotes a unit matrix, Φ denotes the vector of unknown optical field, and A denotes the transverse differential operator. For obtaining the entries of the transverse operator to the e th subdomain, we substitute Eq. (4) with a suitable basis function into Eqs. (2) and (3) and obtain

Alke=Ck(2)(xl)+k02nyy2(xl)δlk,(l,k=0,1,2,...n)
(9)

for TE and that

Alke=nxx2(xl)nzz2(xl)[Ck(2)(xl)2nzz(xl)Ck(1)(xl)]+k02nxx2(xl)δlk,(l,k=0,1,2,...n)
(10)

for TM polarizations. In Eqs. (9) and (10), Ck(n)(xl ) means the n th derivative result of Ck (x) with respect to x at collocation point xl . The global transverse operator A can be obtained by assembling the entries of the transverse operators to all subdomains. However, in matrix A, the rows located in the dielectric interfaces xr ({r=1, 2,…m}, where m denotes the number of discontinuous nodes) shared by two adjacent subdomains are imposed to be replaced by the interfacial boundary conditions

Ey(xr+)=Ey(xr),Ey(xr+)x=Ey(xr)x
(11)

for TE and that

Hy(xl+)=Hy(xl),nzz2(xr)Hy(xr+)x=nzz2(xr+)Hy(xr)x
(12)

for TM polarizations, where the xr+ and xr are, respectively, referred to the locations at the infinitesimal right and left of the interface xr .

3.2 Determining basis functions and boundary conditions

The best choice is that the interior and outermost subdomains are expanded, respectively, by the Chebyshev polynomials and LG functions, if only guided modes exist [20–22

20. C. C. Huang, C. C. Huang, and J. Y. Yang, “An efficient method for computing optical waveguides with discontinuous refractive index profiles using spectral collocation method with domain decomposition,” J. Lightwave Technol. 21, 2284–2296 (2003). [CrossRef]

]. However, the outermost subdomains represented by the LG functions are inappropriate while tackling the waveguide structures with leaky modes. In this work, we mainly propose an efficient solution method for handling leaky waveguides. In general, the optical fields in the outermost subdomains exhibit two conditions: one is with evanescent wave, and the other is with traveling wave. Certainly, the outermost subdomains with the evanescent decay characteristic can be efficiently represented by the LG functions as well as those considered in Refs. [20–22

20. C. C. Huang, C. C. Huang, and J. Y. Yang, “An efficient method for computing optical waveguides with discontinuous refractive index profiles using spectral collocation method with domain decomposition,” J. Lightwave Technol. 21, 2284–2296 (2003). [CrossRef]

]. As for that with traveling field, since the oscillatory behavior resulting from leaky modes penetrating to substrate, which have a larger refractive index than that in the core layer, this kind of subdomain is expanded by the Chebyshev polynomials at collocation points but has to be fulfilled by the Mur’s ABC [23

23. G. Mur, “Absorbing boundary conditions for the finite difference approximation of the time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. 23, 377–382 (1981). [CrossRef]

] at the boundary point (i. e., outmost collocation point), which allows the fields to radiate freely out of the computational window. The perfectly matched layer (PML) condition may be a more efficient scheme than Mur’s ABC, but the implementation of PML to the present method is very complicated. In this paper, we mainly aim to validate the accuracy and efficiency of the pseudospectral approach to leaky mode problems. As a result, we prefer choosing a simpler boundary condition like Mur’s ABC to accomplish the work.

In this work, we adopt the second-order formula of Mur’s schemes in frequency domain, and it can be given as follows:

(r+ikr)2φ=0,
(13)

where i=1 , φ denotes the unknown field and is expanded by the Chebyshev polynomials, r denotes the outward direction normal to the boundary, and kr denotes the wavenumber in r-direction. In the paper, r and kr can be replaced by x and kx in the present coordinate system, respectively. Further, Eq. (13) can be expanded as follows:

(2x2+2ikxxkx2)φx=xbp=0,
(14)

where xbp denotes the boundary point in the outermost subdomain. Note that the chosen computational interval of the outermost subdomain expanded by the Chebyshev polynomials slightly affects the convergence. Observing Eq. (14), we find that the plane wave solution of φ(x)=φ(0)exp(-ikxx)is fulfilled [17

17. H. P. Uranus, H. J. W. M. Hoekstra, and E. V. Groesen, “Simple high-order Galerkin finite scheme for the investigation of both guided and leaky modes in anisotropic planar waveguides,” Opt. Quantum Electron. 36, 239–257 (2004). [CrossRef]

]. After that, we substitute the plane wave solution into Eqs. (2) and (3) for the outermost subdomain with homogeneous refractive index; we have

kx=k0nyy2neff2
(15)

for TE, and

kx=k0nzznxxnxx2neff2,
(16)

for TM polarizations. Through Eqs. (2) and (3), the first term in Eq. (14) can be replaced by the terms of -kx2 φ for both TE and TM polarizations. As a result, Eq. (14) can be further modified as follows:

(2ikxx2kx2)φx=xbp=0
(17)

Besides the consideration of choosing basis functions in the present work, the Chebyshev polynomials with Mur’s ABC can also be used to represent the evanescent field having decaying characteristics at the boundary as well as that used in the finite element scheme with Sommerfeld’s ABC [17

17. H. P. Uranus, H. J. W. M. Hoekstra, and E. V. Groesen, “Simple high-order Galerkin finite scheme for the investigation of both guided and leaky modes in anisotropic planar waveguides,” Opt. Quantum Electron. 36, 239–257 (2004). [CrossRef]

]. However, in our numerical experiments concerning computational efficiency (the required number of basis functions), the preferable choice is still the LG functions. Moreover, no supplementary boundary conditions are demanded for evanescent fields [20–22

20. C. C. Huang, C. C. Huang, and J. Y. Yang, “An efficient method for computing optical waveguides with discontinuous refractive index profiles using spectral collocation method with domain decomposition,” J. Lightwave Technol. 21, 2284–2296 (2003). [CrossRef]

], and thus the numerical implementation may be significantly simplified.

4. Numerical results and discussion

In order to verify the performances of the present scheme as applied to various ARROW structures, we first analyze in detail the propagation characteristics of an isotropic example, which has been studied by Baba and Kokubun [6

6. T. Baba and Y. Kokubun, “Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides-numerical results and analytical expressions,” IEEE J. Quantum Electron. 28, 1689–1700 (1992). [CrossRef]

] using the system interference matrix and by Chen et al. [14

14. C. K. Chen, P. Berini, D. Feng, S. Tanev, and V. P. Tzolov, “Efficient and accurate numerical analysis of multilayer planar waveguides in lossy anisotropic media,” Opt. Express 7, 260–272 (2000). [CrossRef] [PubMed]

] using TMM with APM. Except for the anisotropy, the second example is the same as the first example. Finally, we study the coupling length and field distributions of even and odd modes for an ARROW-based directional coupler device. The accuracy of these examples calculated by the proposed approach is discussed and compared with other published results.

4.1 Isotropic ARROW structure

The schematic diagrams of the geometry and refractive index profile of the ARROW structure are depicted in Fig. 1. The relevant parameters used are those with an operating wavelength of λ=0.6328µm, the refractive indices are na=1for air and ns=3.85 for substrate, and the thicknesses (refractive indices) for the first cladding, second cladding, and core layers are, respectively, d1=0.142λ (n1=2.3), d2=3.15λ (n2=1.46), and dc=6.3λ (nc=1.46) under the antiresonant condition. The computational domain is divided into five subdomains. The calculated results of the complex effective indices for the first i time iterations neffi are shown in Table 1, which also shows that neff0=1.459 denotes the chosen initial guess of effective index. In the example, the number of terms of LG basis functions for the outermost subdomain occupied by air is Na=10 while the computational interval estimated according to our criterion [21

21. C. C. Huang, C. C. Huang, and J. Y. Yang, “A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles,” IEEE J. Sel. Top. Quantum Electron. 11, 457–465 (2005). [CrossRef]

] for spreading of guided mode is Sa =1µm, the number of Chebyshev polynomials for the first cladding, second cladding, and core layers are N1=N2=Nc=20, and the Chebyshev polynomials for substrate layer is Ns=40 while the computational interval of the outermost subdomain chosen is Ssub =1µm. More terms of basis functions in the substrate layer are necessary due to the sharply oscillatory behavior of leaky waves. In addition, considering the effect caused by the computational interval of the outermost subdomain with higher refractive index, the differences of the convergent results between Ssub =1µm and Ssub =2µm for all of the modes are negligible while using Ns=50 for Ssub =2µm. We conclude that the major factor in obtaining accurate results is use of the pseudospectral scheme. Table 1 also shows that with only three iterations, our method achieves the convergent values. In addition, the exact four-digit values for neff can be obtained even when only one iteration is executed. In our numerical experiments, the present approach obtains the same convergent rate for using different neff0, as long as the chosen value of neff0 is slightly smaller than nc. The solutions of this work, obtained by the third iteration, and the results calculated by Chen et al. [14

14. C. K. Chen, P. Berini, D. Feng, S. Tanev, and V. P. Tzolov, “Efficient and accurate numerical analysis of multilayer planar waveguides in lossy anisotropic media,” Opt. Express 7, 260–272 (2000). [CrossRef] [PubMed]

] are shown in Table 2. We can see that the two schemes are in excellent agreement.

Fig. 1. Schematic diagrams of geometry and refractive index profile of the ARROW structure.

Table 1. Complex Effective Indices of the TE and TM Modes of an Isotropic ARROW Structure by the Present Method for the First Three Iterations

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The initial guess of effective index is neff0=1.459, and the number of terms of basis function in each subdomain is Na=10, N1=N2=Nc=20, and Ns=40.

Table 2. Calculated Results of the Present Method and TMM Using APM [14]

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To further demonstrate the accuracy of our method, we calculate the dispersion and radiation loss characteristics of the ARROW structure versus the thickness of each layer while the thicknesses of other layers are at corresponding antiresonant conditions. First, Figs. 2(a) and 2(b) show the dispersion and loss characteristics of the first six ARROW modes of TE polarization, labeled TE1-TE6, versus the thickness of the first cladding (d1/λ), respectively, while other parameters are fixed. Here, referring to Ref. [6

6. T. Baba and Y. Kokubun, “Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides-numerical results and analytical expressions,” IEEE J. Quantum Electron. 28, 1689–1700 (1992). [CrossRef]

], we call TE0 as the first cladding mode. In Fig. 2(a), the ranges of d1/λ with flat portions of effective index for each mode correspond to the antiresonances of Fabry-Perot cavities formed in the two interference claddings. The same portions represent the low-loss portions while referring to Fig. 2(b). It is clear that the low-loss portion of TE1 is broad so as to allow large fabrication tolerance. In contrast to the flat portions, the transitional portions in Fig. 2(a) illustrate the resonances subject to high-loss portions in Fig. 2(b). For comparing the order of losses between different polarized waves, Fig. 2(b) also illustrates the first two ARROW modes of TM polarization. It is clear that the losses of TM modes are larger than TE ones at a corresponding mode number. As a result, the higher-order TE and TM modes can be easily filtered out to obtain single-mode (TE1) propagation. Besides, in Fig. 2(b), the loss characteristics display periodic variety as a function of (d1/λ). Likewise, the propagation characteristics versus the thickness of the second cladding (d2/λ) and core (dc/λ) layers are shown in Figs. 3 and 4, respectively. In Figs. 3(b) and 4(b), at corresponding first antiresonant conditions, high loss discrimination as that observed in Fig. 2(b) is also shown.

Fig. 2. (a). Dispersion characteristics; (b).Radiation loss characteristics of various modes of an isotropic ARROW versus the thickness of the first cladding.
Fig. 3. (a). Dispersion characteristics; (b). Radiation loss characteristics of various modes of an isotropic ARROW versus the thickness of the second cladding.
Fig. 4. (a). Dispersion characteristics; (b). Radiation loss characteristics of various modes of an isotropic ARROW versus the thickness of the core.

Furthermore, Figs. 5(a)–5(f) show the real part of relative field profiles of various TE (solid curve) and TM (dotted curve) polarized waves simultaneously at the first antiresonant condition (i. e., d1=0.142λ, d2=3.15λ, and dc=6.3λ). Here, “relative” means that the field profiles are normalized by their maximum values [6

6. T. Baba and Y. Kokubun, “Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides-numerical results and analytical expressions,” IEEE J. Quantum Electron. 28, 1689–1700 (1992). [CrossRef]

]. In this paper, Figs. 5(a)–5(f) are sequentially labeled as TE0 (TM0) to TE6 (TM6). We can observe that the fields in Fig. 5(a) are confined to the vicinity of the first cladding layer, and the confinement of the TE mode is tighter than that of the TM mode. Figure 5(b) illustrates that the lowest ARROW mode (TE1) is most localized in the core layer, which is of practical interest, and that it is similar to the fundamental mode of conventional waveguides. The oscillatory characteristics in the substrate layer are also clearly shown, and the larger loss of TM mode is observed, as well as the values listed in Table 2. From Figs. 5(b)–5(f), excluding Figs. 5(b) and 5(e), the fields penetrating the interference claddings are fairly large.

Fig. 5. Relative field profiles of TE and TM modes of isotropic ARROW on the first antiresonant condition on (d1=0.142λ,d2=3.15λ, and dc=6.3λ): (a) TE0 and TM0 (the first cladding mode); (b) TE1 and TM1; (c)TE2 and TM2; (d) TE3 and TM3; (e) TE4 and TM4; (f) TE5 and TM5.

4.2 Anisotropic ARROW structure

Following the same structure analyzed in Subsection 4.1, but with anisotropy in inner layers sandwiched between air and substrate layers, the anisotropic material with a diagonal form of dielectric tensor is expressed in Eq. (1). The refractive indices are nzzc=1.46 for the core, nzz1=2.3for the first cladding, and nzz2=1.46 for the second cladding. Other values follow the relations of nxxj=nyyj=1.03nzzj, where (j=c, 1,2). Here, the initial guess of effective index of neff0=1.459 is chosen, and three iterations are executed to achieve convergent solutions. The results obtained by our method, using 140 unknowns (Na=10,N1=N2=Nc=30,Ns=40), are shown in Table 3 with the six-order accuracy of the finite element method [17

17. H. P. Uranus, H. J. W. M. Hoekstra, and E. V. Groesen, “Simple high-order Galerkin finite scheme for the investigation of both guided and leaky modes in anisotropic planar waveguides,” Opt. Quantum Electron. 36, 239–257 (2004). [CrossRef]

] and TMM using APM [14

14. C. K. Chen, P. Berini, D. Feng, S. Tanev, and V. P. Tzolov, “Efficient and accurate numerical analysis of multilayer planar waveguides in lossy anisotropic media,” Opt. Express 7, 260–272 (2000). [CrossRef] [PubMed]

]. The number of unknowns in Ref. [17

17. H. P. Uranus, H. J. W. M. Hoekstra, and E. V. Groesen, “Simple high-order Galerkin finite scheme for the investigation of both guided and leaky modes in anisotropic planar waveguides,” Opt. Quantum Electron. 36, 239–257 (2004). [CrossRef]

] is nearly 1300 because of the effective mesh size of 0.005µm. This shows that the computational efficiency of the present scheme is superior to that of the finite element method with six-order accuracy [17

17. H. P. Uranus, H. J. W. M. Hoekstra, and E. V. Groesen, “Simple high-order Galerkin finite scheme for the investigation of both guided and leaky modes in anisotropic planar waveguides,” Opt. Quantum Electron. 36, 239–257 (2004). [CrossRef]

]. In addition, the results of complex effective indices obtained by our work show excellent agreement with exact solutions [14

14. C. K. Chen, P. Berini, D. Feng, S. Tanev, and V. P. Tzolov, “Efficient and accurate numerical analysis of multilayer planar waveguides in lossy anisotropic media,” Opt. Express 7, 260–272 (2000). [CrossRef] [PubMed]

]. However, the present scheme obtains a full matrix. The computational time is 0.18 sec (each iteration) executed on Pentium IV PC with a CPU clock rate of 3.0 GHz in a double precision. We obtain that the effective indices and losses as a function of thickness of each inner layer are similar to its isotropic counterpart. As for that the principal dielectric axes of the crystal are not parallel to the waveguide coordinate system, the more complicated hybrid modes and the propagation characteristics depend strongly on the included angle between the optics axis and the waveguide coordinate system.

Table 3. Calculated Results of an Anisotropic ARROW Structure by the Present Method, Six-Order Accuracy FEM [17], and TMM Using APM [14]

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4.3 ARROW-based directional coupler

The directional coupler, which consists of two parallel waveguides, is an interesting photonic structure to build various optical communication devices such as a modulator, filter, power divider, and polarizer. The coupling mechanism of a conventional directional coupler is well studied by the coupled mode theory [24

24. A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3, 1135–1146 (1985). [CrossRef]

], which considers merely the weakly coupled approximation. In addition, the coupling strength decreases as an exponential function of waveguide separation. Because of the widespread applications of ARROW-based structures, a novel dual ARROW-based directional coupler utilizing antiresonant reflection as the guiding mechanism has been proposed to investigate the propagation characteristics [25

25. M. Mann, U. Trutschel, C. Wachter, L. Leine, and F. Lederer, “Directional coupler based on an antiresonant reflecting optical waveguide,” Opt. Lett. 16, 805–807 (1991). [CrossRef] [PubMed]

]. In comparison with a conventional directional coupler that utilizes total internal reflection, the ARROW structures have strong coupling, which is due to their having leaky waves rather than the evanescent waves found in conventional two parallel waveguides; furthermore, the variation of the coupling length versus waveguide separation displays periodicity [25

25. M. Mann, U. Trutschel, C. Wachter, L. Leine, and F. Lederer, “Directional coupler based on an antiresonant reflecting optical waveguide,” Opt. Lett. 16, 805–807 (1991). [CrossRef] [PubMed]

].

In Fig. 6, the configuration of the dual ARROW structure grown on a Si substrate with a refractive index ns=3.5 consists of two ARROW structures separated by a separation cladding layer with a refractive index nsep=1.46 and thickness dsep=2µm [26

26. Y. H. Chen and Y. T. Huang, “Coupling-efficiency analysis and control of dual antiresonant reflecting optical waveguides,” J. Lightwave Technol. 14, 1507–1513 (1996). [CrossRef]

]. At the operating wavelength λ=0.6328µm, the two waveguide cores are with nc1=nc2=1.46 and dc1=dc2=4µm. The core 1 is sandwiched by the two thin cladding layers with nh1=nh2=2.3 and dh1=dh2=0.089µm, and the same circumstance is also encountered by core 2, which is sandwiched by two cladding layers with nh3=nh4=2.3and dh3=dh4=0.089µm. For the upper cladding layer, which is grown over the thin cladding layer of dh1 and in contact with air having a na=1, the parameters are n11=1.46 and d11=2µm. As for the lower cladding layer grown over the Si substrate, it is with n12=1.46 and d12=2µm. The purpose of the different interference claddings is to accomplish the antiresonant conditions. According to the conclusion found by Chen and Huang [26

26. Y. H. Chen and Y. T. Huang, “Coupling-efficiency analysis and control of dual antiresonant reflecting optical waveguides,” J. Lightwave Technol. 14, 1507–1513 (1996). [CrossRef]

], the coupling efficiency can be flexibly controlled through the adjustment of outermost cladding thickness d11. If we consider the conditions of d11=d12=2µm and d11=0,dh1=0.03µm, the maximum coupling efficiency of Co=99.87% (while the coupling length is Lc=λ/(Ns-Na)=59mm, where Ns and Na denote lowest order symmetric (even) and asymmetric (odd) modes, respectively) and decoupled phenomenon can be achieved [26

26. Y. H. Chen and Y. T. Huang, “Coupling-efficiency analysis and control of dual antiresonant reflecting optical waveguides,” J. Lightwave Technol. 14, 1507–1513 (1996). [CrossRef]

], respectively.

Fig. 6. Schematic diagrams of the coupling structure and refractive index profile of the ARROW-based directional coupler with Si substrate, where dg1, dg1, and dsep are the thicknesses of core 1, core 2, and waveguide separation, respectively. Others layers, excluding the half space of air and substrate layers, d11, dh1, dh2 dh3, dh4, and d12 are the interference cladding layers.

To validate the characteristics studied in Ref. [26

26. Y. H. Chen and Y. T. Huang, “Coupling-efficiency analysis and control of dual antiresonant reflecting optical waveguides,” J. Lightwave Technol. 14, 1507–1513 (1996). [CrossRef]

] and demonstrate the ability of our scheme, by the present scheme, the dual ARROW structure is considered as an 11-layer waveguide structure, and the computational window is divided into 11 subdomains. The numbers of terms of LG functions used for air is Na=10, of Chebyshev polynomials for substrate is Ns=40, and of others are Ni=20 (where i represents other layers). Compared with the coupling length of Lc=59mm, obtained in Ref. [26

26. Y. H. Chen and Y. T. Huang, “Coupling-efficiency analysis and control of dual antiresonant reflecting optical waveguides,” J. Lightwave Technol. 14, 1507–1513 (1996). [CrossRef]

], the calculated coupling length is Lc=58.60mm (where Ns=1.45785857 and Na=1.45785317). The coupling length of TM mode obtained by our scheme is Lc=9.51mm (where Ns=1.45787059 and Na=1.45783732), which is far smaller than that of the TE mode. This is because the coupling resonances of leaky waves of TM modes are larger than that of TE modes. In our calculation, the imaginary parts (radiation loss) of complex effective indices of TM modes are two orders larger than that of TE modes (namely, TE:≈10-6 dB/cm, TM:≈10-4 dB/cm). From the dispersion characteristics of TE modes illustrated in Fig. 7 it can be clearly seen that the maximum coupling is located at d11=2µm, making the dual ARROW acts as a symmetric coupler, and the decoupling portions are close to d11=0µm and d11=4µm. Additionally, the field profiles of the even and odd TE modes for three cases of d11=2µm (maximum coupling), d11=0.5µm (half coupling), and d11=0 (decoupling, while dh1=0.03µm is used [26

26. Y. H. Chen and Y. T. Huang, “Coupling-efficiency analysis and control of dual antiresonant reflecting optical waveguides,” J. Lightwave Technol. 14, 1507–1513 (1996). [CrossRef]

]) are clearly shown by the decreased order of coupling in Figs. 8(a)–8(c), respectively. Finally, in Fig. 9, the coupling characteristics of the TE and TM modes as a function of the waveguide separation are also illustrated. Obviously, the coupling lengths of the TE and TM modes appear to be a periodic function, as predicted in Ref. [25

25. M. Mann, U. Trutschel, C. Wachter, L. Leine, and F. Lederer, “Directional coupler based on an antiresonant reflecting optical waveguide,” Opt. Lett. 16, 805–807 (1991). [CrossRef] [PubMed]

]. Consequently, this shows that we may apply a dual ARROW-based coupler to easily accomplish a variety of applications in a remote coupler [25

25. M. Mann, U. Trutschel, C. Wachter, L. Leine, and F. Lederer, “Directional coupler based on an antiresonant reflecting optical waveguide,” Opt. Lett. 16, 805–807 (1991). [CrossRef] [PubMed]

].

Fig. 7. Dispersion characteristics versus the upper cladding layer d11 for the lowest symmetric (even) and asymmetric (odd) modes.
Fig. 8. Relative field profiles of the lowest symmetric (even) and asymmetric (odd) modes at different order of coupling: (a) maximum coupling, (b) half coupling, (c) decoupling.
Fig. 9. The coupling lengths of TE and TM modes versus waveguide separation dsep.

5. Conclusion

We successfully demonstrate the ability of the present method to deal with the leaky modes supported by the isotropic and anisotropic ARROW structures. The main consideration is to represent the outermost subdomains with strongly oscillatory characteristics by the efficient Chebyshev polynomials with Mur’s ABC allowing outgoing waves to freely penetrate into substrate, evading most reflected waves. As a result, the oscillatory characteristics are accurately represented. The results calculated by the present scheme show excellent agreement with exact solutions and the computational efficiency is far higher than the finite element method. The application of our scheme to a dual ARROW structure is considered as an ARROW-based directional coupler; the accurate values of the coupling length are obtained, and periodicity of the coupling length versus waveguide separation is also validated. The extension of the present method to three-dimensional ARROW structures of practical interests will be reported elsewhere.

Acknowledgments

This work was supported by the National Science Council, Taiwan, Republic of China under contract No. NSC 95-2221-E-275-005.

References and links

1.

M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49, 13–15 (1986). [CrossRef]

2.

T. Baba, Y. Kokubun, T. Sakaki, and K. Iga, “Loss reduction of an ARROW waveguide in shorter wavelength and its stack configuration,” J. Lightwave Technol. 6, 1440–1445 (1988). [CrossRef]

3.

M. Mann, U. Trutschel, C. Wachter, L. Leine, and F. Lederer, “Directional coupler based on antiresonant reflecting optical waveguide,” Opt. Lett. 16, 805–807 (1991). [CrossRef] [PubMed]

4.

Z. M. Mao and W. P. Huang, “An ARROW optical wavelength filter: design and analysis,” J. Lightwave Technol. 11, 1183–1188 (1992). [CrossRef]

5.

F. Prieto, A. Llobera, D. Jimenez, C. Domenguez, A. Calle, and L. M. Lechuga, “Design and analysis of silicon antiresonant reflecting optical waveguides for evanescent field sensor,” J. Lightwave Technol. 18, 966–972 (2000). [CrossRef]

6.

T. Baba and Y. Kokubun, “Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides-numerical results and analytical expressions,” IEEE J. Quantum Electron. 28, 1689–1700 (1992). [CrossRef]

7.

W. P. Huang, R. M. Shubair, A. Nathan, and Y. L. Chow, “The modal characteristics of ARROW structures,” J. Lightwave Technol. 10, 1015–1022 (1992). [CrossRef]

8.

T. Baba and Y. Kokubun, “New polarization-insensitive antiresonant reflecting optical waveguide (ARROW-B),” IEEE Photon. Technol. Lett. 1, 232–234 (1989). [CrossRef]

9.

W. Jiang, J. Chrostowski, and M. Fontaine, “Analysis of ARROW waveguides,” Opt. Commun. 72, 180–186 (1989). [CrossRef]

10.

J. Chilwell and I. Hodgkinson, “Thin-film field-transfer matrix theory for planar multilayer waveguides and reflection from prism-loaded waveguides,” J. Opt. Soc. Am. A 1, 742–753 (1984). [CrossRef]

11.

J. Kubica, D. Uttamchandani, and B. Culshaw, “Modal propagation within ARROW waveguides,” Opt. Commun. 78, 133–136 (1990). [CrossRef]

12.

J. M. Kubica, “Numerical analysis of InP/InGaAsP ARROW waveguides using transfer matrix approach,” J. Lightwave Technol. 10, 767–771 (1992). [CrossRef]

13.

E. Anemogiannis and E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344–1351 (1992). [CrossRef]

14.

C. K. Chen, P. Berini, D. Feng, S. Tanev, and V. P. Tzolov, “Efficient and accurate numerical analysis of multilayer planar waveguides in lossy anisotropic media,” Opt. Express 7, 260–272 (2000). [CrossRef] [PubMed]

15.

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer boundary conditions for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8, 652–654 (1996). [CrossRef]

16.

J. C. Grant, J. C. Beal, and N. J. P. Frenette, “Finite element analysis of the ARROW leaky optical waveguide,” IEEE J. Quantum Electron. 30, 1250–1253 (1994). [CrossRef]

17.

H. P. Uranus, H. J. W. M. Hoekstra, and E. V. Groesen, “Simple high-order Galerkin finite scheme for the investigation of both guided and leaky modes in anisotropic planar waveguides,” Opt. Quantum Electron. 36, 239–257 (2004). [CrossRef]

18.

Y. Tsuji and M. Koshiba, “Guided-mode and leaky-mode analysis by imaginary distance beam propagation method based on finite element scheme,” J. Lightwave Technol. 18, 618–623 (2000). [CrossRef]

19.

J. P. Boyd, “Chebyshev and Fourier Spectral methods,” in Lecture Notes in Engineering, 2nd ed. (Springer Verlag, 2001).

20.

C. C. Huang, C. C. Huang, and J. Y. Yang, “An efficient method for computing optical waveguides with discontinuous refractive index profiles using spectral collocation method with domain decomposition,” J. Lightwave Technol. 21, 2284–2296 (2003). [CrossRef]

21.

C. C. Huang, C. C. Huang, and J. Y. Yang, “A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles,” IEEE J. Sel. Top. Quantum Electron. 11, 457–465 (2005). [CrossRef]

22.

C. C. Huang and C. C. Huang, “An efficient and accurate semivectorial spectral collocation method for analyzing polarized modes of rib waveguides,” J. Lightwave Technol. 23, 2309–2317 (2005). [CrossRef]

23.

G. Mur, “Absorbing boundary conditions for the finite difference approximation of the time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. 23, 377–382 (1981). [CrossRef]

24.

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3, 1135–1146 (1985). [CrossRef]

25.

M. Mann, U. Trutschel, C. Wachter, L. Leine, and F. Lederer, “Directional coupler based on an antiresonant reflecting optical waveguide,” Opt. Lett. 16, 805–807 (1991). [CrossRef] [PubMed]

26.

Y. H. Chen and Y. T. Huang, “Coupling-efficiency analysis and control of dual antiresonant reflecting optical waveguides,” J. Lightwave Technol. 14, 1507–1513 (1996). [CrossRef]

OCIS Codes
(130.2790) Integrated optics : Guided waves
(230.7390) Optical devices : Waveguides, planar

ToC Category:
Integrated Optics

History
Original Manuscript: September 18, 2006
Revised Manuscript: October 23, 2006
Manuscript Accepted: October 31, 2006
Published: November 27, 2006

Citation
Chia-Chien Huang, "Numerical calculations of ARROW structures by pseudospectral approach with Mur’s absorbing boundary conditions," Opt. Express 14, 11631-11652 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-24-11631


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References

  1. M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, "Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures," Appl. Phys. Lett. 49, 13-15 (1986). [CrossRef]
  2. T. Baba, Y. Kokubun, T. Sakaki, and K. Iga, "Loss reduction of an ARROW waveguide in shorter wavelength and its stack configuration," J. Lightwave Technol. 6, 1440-1445 (1988). [CrossRef]
  3. M. Mann, U. Trutschel, C. Wachter, L. Leine, and F. Lederer, "Directional coupler based on antiresonant reflecting optical waveguide," Opt. Lett. 16, 805-807 (1991). [CrossRef] [PubMed]
  4. Z. M. Mao and W. P. Huang, "An ARROW optical wavelength filter: design and analysis," J. Lightwave Technol. 11, 1183-1188 (1992). [CrossRef]
  5. F. Prieto, A. Llobera, D. Jimenez, C. Domenguez, A. Calle, and L. M. Lechuga, "Design and analysis of silicon antiresonant reflecting optical waveguides for evanescent field sensor," J. Lightwave Technol. 18, 966-972 (2000). [CrossRef]
  6. T. Baba and Y. Kokubun, "Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides-numerical results and analytical expressions," IEEE J. Quantum Electron. 28, 1689-1700 (1992). [CrossRef]
  7. W. P. Huang, R. M. Shubair, A. Nathan, and Y. L. Chow, "The modal characteristics of ARROW structures," J. Lightwave Technol. 10, 1015-1022 (1992). [CrossRef]
  8. T. Baba and Y. Kokubun, "New polarization-insensitive antiresonant reflecting optical waveguide (ARROW-B)," IEEE Photon. Technol. Lett. 1, 232-234 (1989). [CrossRef]
  9. W. Jiang, J. Chrostowski, and M. Fontaine, "Analysis of ARROW waveguides," Opt. Commun. 72, 180-186 (1989). [CrossRef]
  10. J. Chilwell and I. Hodgkinson, "Thin-film field-transfer matrix theory for planar multilayer waveguides and reflection from prism-loaded waveguides," J. Opt. Soc. Am. A 1, 742-753 (1984). [CrossRef]
  11. J. Kubica, D. Uttamchandani, and B. Culshaw, "Modal propagation within ARROW waveguides," Opt. Commun. 78, 133-136 (1990). [CrossRef]
  12. J. M. Kubica, "Numerical analysis of InP/InGaAsP ARROW waveguides using transfer matrix approach," J. Lightwave Technol. 10, 767-771 (1992). [CrossRef]
  13. E. Anemogiannis and E. N. Glytsis, "Multilayer waveguides: efficient numerical analysis of general structures," J. Lightwave Technol. 10, 1344-1351 (1992). [CrossRef]
  14. C. K. Chen, P. Berini, D. Feng, S. Tanev, and V. P. Tzolov, "Efficient and accurate numerical analysis of multilayer planar waveguides in lossy anisotropic media," Opt. Express 7, 260-272 (2000). [CrossRef] [PubMed]
  15. W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, "The perfectly matched layer boundary conditions for modal analysis of optical waveguides: leaky mode calculations," IEEE Photon. Technol. Lett. 8, 652-654 (1996). [CrossRef]
  16. J. C. Grant, J. C. Beal, and N. J. P. Frenette, "Finite element analysis of the ARROW leaky optical waveguide," IEEE J. Quantum Electron. 30, 1250-1253 (1994). [CrossRef]
  17. H. P. Uranus, H. J. W. M. Hoekstra, and E. V. Groesen, "Simple high-order Galerkin finite scheme for the investigation of both guided and leaky modes in anisotropic planar waveguides," Opt. Quantum Electron. 36, 239-257 (2004). [CrossRef]
  18. Y. Tsuji and M. Koshiba, "Guided-mode and leaky-mode analysis by imaginary distance beam propagation method based on finite element scheme," J. Lightwave Technol. 18, 618-623 (2000). [CrossRef]
  19. J. P. Boyd, "Chebyshev and Fourier Spectral methods," in Lecture Notes in Engineering, 2nd ed. (Springer Verlag, 2001).
  20. C. C. Huang, C. C. Huang, and J. Y. Yang, "An efficient method for computing optical waveguides with discontinuous refractive index profiles using spectral collocation method with domain decomposition," J. Lightwave Technol. 21, 2284-2296 (2003). [CrossRef]
  21. C. C. Huang, C. C. Huang, and J. Y. Yang, "A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles," IEEE J. Sel. Top. Quantum Electron. 11, 457-465 (2005). [CrossRef]
  22. C. C. Huang and C. C. Huang, "An efficient and accurate semivectorial spectral collocation method for analyzing polarized modes of rib waveguides," J. Lightwave Technol. 23, 2309-2317 (2005). [CrossRef]
  23. G. Mur, "Absorbing boundary conditions for the finite difference approximation of the time-domain electromagnetic field equations," IEEE Trans. Electromagn. Compat. 23, 377-382 (1981). [CrossRef]
  24. A. Hardy and W. Streifer, "Coupled mode theory of parallel waveguides," J. Lightwave Technol. 3, 1135-1146 (1985). [CrossRef]
  25. M. Mann, U. Trutschel, C. Wachter, L. Leine, and F. Lederer, "Directional coupler based on an antiresonant reflecting optical waveguide," Opt. Lett. 16, 805-807 (1991). [CrossRef] [PubMed]
  26. Y. H. Chen and Y. T. Huang, "Coupling-efficiency analysis and control of dual antiresonant reflecting optical waveguides," J. Lightwave Technol. 14, 1507-1513 (1996). [CrossRef]

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